Editorial

Introduction: Two Kinds of Proportion

Author:

Matthew A Cohen

Washington State University, US

Abstract

The subject of architectural proportional systems in the history of architecture, the topic of this special collection of essays in Architectural Histories, has long been characterized by a fundamental ambiguity: the word and concept of proportion simultaneously signify two unrelated and in some ways opposite meanings. Proportion can refer to ratios, or it can refer to architectural beauty. In this introduction to the papers that follow, Matthew A. Cohen proposes a simple clarification of this ambiguity as a framework for continued discussion of this subject: that whenever scholars use the word proportion, they specify whether they intend ‘proportion-as-ratio’ or ‘proportion-as-beauty’.

The frequent blending of these meanings today, Cohen argues, is a survival of attitudes toward proportional systems in architecture that were prevalent as long ago as the early Renaissance. Cohen proposes an alternative to Rudolf Wittkower's paradigmatic ‘break-away’ theory of the history of proportional systems, according to which virtually everyone accepted proportional systems as sources of universal beauty in architecture until the mid-eighteenth century, and after that time virtually everyone believed that beauty and proportional systems were matters of individual preference. Rather than a long period respectful of tradition followed by a long period skeptical of it, Cohen argues, based in part on a new interpretation of Claude Perrault’s 1683 codification of the notion of positive beauty, that architects and others have always had access to two parallel strands of thought pertaining to proportional systems: a skeptical-pragmatic strand and a respectful-metaphysical strand. This new historical and historiographical interpretation of the problem of architectural proportional systems, and the new vocabulary with which to discuss it critically presented herein, helps to separate aesthetic from historical considerations.

Introduction

The diverse collection of essays presented in this special collection of
Architectural Histories, written by leading scholars in the
field of architectural history, grew out of the international conference
‘Proportional Systems in the History of Architecture’, held in Leiden in
March 2011 (Fig. 1).1 The conference was scheduled to commemorate the sixtieth
anniversary of the last international conference on proportional systems in the
arts, held in Milan in 1951 and titled ‘De divina proportione’, which
similarly gathered leading thinkers of its day (Fig. 2).2 This recent anniversary thus
offers a valuable opportunity to reflect on where the study of proportional systems
has gone over the past sixty years, and where it might most productively go from
here. Although the premises of the two conferences were fundamentally different from
one another — the Milan conference promoted the contemporary use of
proportional systems in the arts for the aesthetic and spiritual betterment of
society, while the Leiden conference promoted the historical study of specifically
architectural proportional systems for the advancement of scholarly knowledge
— certain noteworthy attitudes toward the subject of proportional systems
manifested in the Milan conference are still prominent today.3 On the bright side, both conferences together demonstrate a
sustained recognition of the importance of the multidisciplinary study of
proportional systems as integral parts of human culture across time and geography.
Less productively, while sympathy with the overtly mystical beliefs that drove the
Milan conference is substantially more subdued in the scholarly community today, a
fundamental ambiguity inherent in the concept of proportion that enabled those
beliefs to flourish in 1951 continues to characterize much scholarly thinking about
this subject today: when architectural historians use the word
‘proportion’, whether they intend it to signify a ratio, architectural
beauty, or both simultaneously, is often unclear to author and reader alike.

Fig. 1

Matthew A. Cohen (left) with Mark Wilson Jones (right), at the conference
‘Proportional Systems in the History of Architecture’, Leiden,
17 March 2011. Photo: anonymous gift.

Fig. 2

Rudolf Wittkower addressing the Milan conference ‘De divina
proportione’, as part of the Ninth Triennale, Milan, September 1951.
By permission of Fondazione La Triennale di Milano.

In this introduction I will explore this ambiguity, and propose a clarification of it
to serve as the common thread tying together the two editorial premises of this
special collection of essays: first, that there is no causal relationship between
proportional systems and the aesthetic qualities of architecture; and second, that
proportional systems, as non-visual bearers of meaning and objects of belief,
contributed to the rhetorical rather than visual structure of architecture prior to
the advent of modern structural engineering, which Rowland Mainstone (1968: 303) dates to 1742–1743.4 Proportional systems during this long period may
thus be understood to have served no practical or visual purposes, but nevertheless
to have played critical roles in distinguishing architecture from mere
building.5 After 1742–1743 a limited,
nostalgic strain of pre-engineering, or what I call ‘belief based’,
proportional systems (such as Le Corbusier’s Modulor and Hans van der
Laan’s ‘plastic number’), continued this non-practical tradition
into the 20th century, ignoring the appearance of what I call
‘certainty-based’ proportional systems (such as structural engineering
specifications and urban building regulations), which reflect modern scientific
thinking and succeed in fulfilling practical purposes.6

I do not expect that all of our contributors or readers will necessarily agree with
these premises, but hope that these premises will inspire productive discussion and
debate. Indeed, these premises can be difficult to understand and controversial
because they contradict a set of assumptions so widespread and deeply ingrained in
contemporary thought as to constitute a paradigm.7 I call it the ‘Wittkower paradigm’, in acknowledgement of
Rudolf Wittkower’s critical role in promulgating it by synthesizing various
strands of 19th- and early 20th-century thought in a series of influential
publications after World War II (Wittkower
1949; Wittkower 1953; Wittkower 1960; Cohen 2013: 36–51).8 Paradigms
are not easily overturned, but in this introduction I will at least confront this
one, in order to propose some guidelines to help navigate the inherent ambiguities
of this subject. I will continue this process in the conclusion to this special
collection, where I present ten principles derived from the essays gathered here and
other sources, to serve as a proposed framework for future discussions.

The beauty problem

During my numerous conversations between sessions at the Leiden conference I found
that many of the conference participants tacitly believed, quite strongly, that
proportional systems contribute beauty to architecture. Indeed, as far as I have
been able to determine there and in other venues, such beliefs are held, at least to
some degree, by an overwhelming majority of scholars and architects today.9 Evidently these beauty-in-proportion believers
believe that beauty generated by proportional systems emanates from great buildings
of the past, causing all people to experience visual aesthetic pleasure. Indeed,
many of these believers also believe that scientific principles underlying such
beauty-causing emanations will soon be identified through future studies involving
psychologists, neuroscientists and brain-scanning technology.10 The most recent evidence of such beliefs in a scholarly
publication that I am aware of are the comments by Wittkower from 1960, quoted
below. This belief thus seems to be rather casual and not fully worked out among
scholars today, active in thought and conversation but not publication, though it
remains widespread and influential in some situations.11

Productive discussion of the historical issues pertaining to proportional systems is
today often limited by a general lack of consensus over basic assumptions about the
relationship, or lack thereof, between proportional systems and architectural
beauty. This lack of consensus can have important negative consequences for the
study of architectural history. It can create, for example, what I have termed
‘us/them ambiguity’, which occurs when the belief that proportional
systems create beauty in architecture directs scholarly attention toward
us — i.e., toward our perceptions today — rather
than toward them, or, the people in history whose products,
activities and beliefs are ostensibly the subjects of architectural history (Cohen 2013: 24). Us and
them cannot be considered unified when dealing with the subject
of beauty because assessments of beauty are not universal across time and geography,
as beauty-in-proportion believers assume, but always subjective. Us/them ambiguity
thus creates uncertainty as to whether an investigation is a work of architectural
criticism or history; a commentary on current, or on past, interpretations of
architecture.

To refute the notion that proportional systems have visible — and invariably
favorable — aesthetic influences on architecture, I have previously presented
a logical, five-point argument (Cohen 2013:
281–287). For example, point number one: proportional systems are
mental, not visual constructs. Thus, you cannot see the numerical ratios expressed
in terms of the local unit of measure in use at the time a particular proportional
system was designed. You can only recognize these ratios and their historical
significance after studying measurement data, the history of local units of measure,
and other related information.12 That which is
not visible, therefore, cannot be visually beautiful.13

Sometimes you can see geometrical relationships, but only approximately. For example,
you cannot distinguish between a root-2 rectangle and a slightly stretched one,
though you may think you can (Fig. 3); and in
any case, why should a geometrical figure for which we have a name be more visually
valued than figures for which we have no names?; a question that invokes point
number three: proportional relationships have no intrinsic beauty.14 Thus, a root-2 rectangle cannot have more
intrinsic beauty than a slightly stretched one, because neither of them have any
beauty at all. It follows, therefore, even from this abbreviated argument, that a
proportional system per se, which is but a set of proportional
relationships, cannot contribute beauty to architecture.15

Fig. 3

Summary of the San Lorenzo nave arcade bay proportional system (spread
between two drawings for clarity). Source: author.

I do not aim to spoil anyone’s personal aesthetic experiences of architecture,
nor do I have any illusions that in one essay I could ever disrupt a
beauty-in-proportion belief system that is at least three centuries old. Indeed, the
beauty-in-proportion belief system can co-exist with rigorous scholarship, as long
as the two are kept separated. In this introduction I will propose what such a
separation might look like, first by continuing to shed a critical light on the
beauty-in-proportion belief system, its modern dissemination in particular in the
work of Wittkower, and its origins in the conceptual ambiguity built into the very
word and concept of proportion at least since the 15th century. Second, I will
propose that the word proportion be broken down into its incongruent component
meanings, ‘proportion-as-ratio’ and ‘proportion-as-beauty’,
and that one of these meanings hereafter be specified whenever scholars use this
word, either through the use of the preceding terms or in the context of the
discussion.

One example of the ambiguity that can result when the two meanings of the word
proportion are not separated is found in a published address by Nikolaus Pevsner to
a meeting of the Royal Institute of British Architects held in 1957. When Pevsner
refers to ‘laws of proportions’, ‘fixed proportions’ and
‘proportional canons’, he is clearly referring to proportions-as-ratio;
but when later in the same address he contends that the west front of St.
Paul’s Cathedral ‘is without doubt badly bungled in its
proportions’, he now uses the same word, proportion, to refer to
proportion-as-beauty, and thus reveals his assumption that some particular selection
of proportional ratios caused the west front to lack beauty (Pevsner 1957:
456–457). Pevsner, however, provides no exposition of which particular ratios
he finds offensive. His belief that the west front is unbeautiful due to its
proportions is perhaps metaphysical (for example, a belief that the assumed
proportional ratios in the west front fail to conform to some indistinct notion of
assumed ideal ratios), or perhaps empathetic (a seemingly felt
comparison between the west front and the human body and its states, as in the
towers seeming too large for the nave, like arms too large for a body), or perhaps
some ambiguous combination of the two. Either way, his confusion between imagined
ratios and lack of beauty could have been avoided had he simply noted that he found
the west front to be unbeautiful (or ‘badly bungled’), without any
unqualified reference to proportions. Separating the two meanings of proportion,
alternatively, would have forced Pevsner to confront his beauty-in-proportion
preconceptions, and made his criticism of the west front more acute.16

Before exploring in more detail the double meaning of the word proportion, I need to
clarify my use of another multivalent word in this introduction,
‘aesthetic’. Mostly I will use this term to refer to judgments of beauty
based on relationships between sense perception and taste.17 In architecture, the sense perceptions of primary concern
are usually visual. This use of the term aesthetic with an emphasis on sense
perception was appropriated into German from the Greek aisthesis
(meaning ‘perception’ or ‘sensation’) in 1750 by Alexander
Gottlieb Baumgarten, who considered aesthetics to be a ‘science of perception
that is acquired by means of the senses’ (Harrison, Wood and Gaiger 2000: 489).18 Since the 19th century the term has typically carried the more
general meaning: ‘Of or pertaining to the appreciation or criticism of the
beautiful’, whether manmade or natural.19
Aesthetics, however, can also refer to theories of what beauty is and how it arises.
Finally, aesthetics can refer to the philosophical discipline that explores the
cognitive basis of beauty, with fundamental texts by Baumgarten, Kant, Hegel,
Schelling and others. I find Baumgarten’s insistence on sense perception as a
central concern of aesthetics, if not his conception of aesthetics as a science, to
be a useful tool for grounding discussions of the question of causation between
proportional systems and architectural beauty, because it directs our attention
toward the physical properties of architecture and away from vague notions of
emotional satisfaction that could have any cause, related to architecture or
not.

The ‘beauty problem’ discussed here may indeed be that many scholars
today are to varying degrees distracted from important areas of scholarly inquiry by
the belief that proportional systems contribute beauty to architecture, but some
readers may now be wondering where beauty in architecture — at least of the
orderly-looking variety — comes from, if not proportional systems. The answer
to this question will not be found in this special collection of essays, the focus
of which is architectural history rather than criticism. Indeed, no definitive
answer will ever be found, because beautiful, orderly-looking buildings can only be
so when someone believes them to be. I find education and empathy theory to provide
more useful explanations for architectural beauty than proportional-system
metaphysics, but this opinion requires a different venue for discussion.20

The Wittkower paradigm

The modern study of proportional systems in the history of architecture may be said
to have started with the publications of James S. Ackerman’s (1949) ‘“Ars sine scientia nihil
est”: Gothic Theory of Architecture at the Cathedral of Milan’, and
Wittkower’s (1949)
Architectural Principles in the Age of Humanism. Both of these
works demonstrated that valuable insights into the meanings associated with
proportional systems by architects and other thinkers of the past could be derived
from the scholarly study of historical texts. Prior to these publications, studies
of architectural proportions-as-ratio were almost exclusively aesthetic, attempting
to determine why Gothic and classical architecture looks the way it does either
through geometrical and modular reconstructions, or through mystical
ruminations.21 Although Ackerman’s
study remains a landmark in the scholarship of Gothic architectural theory and
practice, it has remained singular because of the uniqueness of the Milan archives
on which it is closely based. Wittkower’s study, by contrast, with its broad
theoretical generalizations and citations from a wide variety of primary sources,
created a new school of thought pertaining to proportional systems in the medieval
and Renaissance periods.22

Despite Wittkower’s stated intention in publishing Architectural
Principles, ‘“to dispose, once and for all, of the
hedonist, or purely aesthetic, theory of Renaissance architecture”’, he
instead replaced one aesthetic theory with another.23 Indeed, the word ‘purely’ in this passage suggests that
he did not object to all aesthetic interpretations of Renaissance architecture, but
only those that treated it as ‘art-as-such’, independent of any
theoretical, social, practical or other considerations.24 For Wittkower, architectural aesthetics contained meanings as well as
feelings. His notion of aesthetics is thus more inclusive than Immanuel Kant’s
definition of ‘Taste as the faculty of judging of that which makes universally
communicable, without the mediation of a concept, our feeling in a given
representation’ (Kant 1914: 173).25 For Wittkower, a mediating concept is
necessary for any notion of aesthetics worthy of a thinking person’s
attention.26

The most easily recognizable component of the Wittkower paradigm is the theory that I
call ‘medieval geometry vs. Renaissance number’.27 It is based on two premises, one historical and the other
aesthetic. Wittkower (1953: 15) summarizes
the historical premise as follows: ‘two different classes of proportion, both
derived from the Pythagoreo-Platonic world of ideas, were used during the long
history of European art, and […] the Middle Ages favored […] geometry,
while the Renaissance and classical periods preferred the numerical, i.e. the
arithmetical side of the tradition’. Wittkower, however, is not the originator
of this premise. As early as 1867, Joseph Gwilt, in his popular Encyclopedia
of Architecture, had already characterized the history of Western
architectural theory as a contrast between geometrical ‘medieval
proportion’ and numerical or grid-based classical architectural proportion,
and Gwilt had merely compiled these notions from earlier sources.28 Wittkower infused this 19th-century formulation, which may
have originally carried implicit aesthetic undertones, with an explicitly aesthetic
premise by associating it with Alois Riegl’s concept of
Kunstwollen (Cohen (2013:
47).29 Thus he claims that
‘the Renaissance attitude to proportion was determined by a new organic
approach to nature, which involved the empirical procedure of measuring, and was
aimed at demonstrating that everything was related to everything by number’
(Wittkower 1953: 16). By contrast, he
claims, ‘the mediaeval quest for ultimate truth behind appearances was
perfectly answered by geometrical configurations of a decisively fundamental nature;
that is, by geometrical forms which were irreconcilable with the organic structure
of figure and building’ (Wittkower 1953:
17). Thus according to Wittkower, the kinds of proportional systems used
in each period can be interpreted as aesthetic reflections of the worldview of each
period.

The historical premise of this theory, considered independently of the aesthetic
premise, at best represents an oversimplification of available evidence, which
indicates that during both the medieval and Renaissance periods geometry and number
served as equal partners in architectural theory and practice (Cohen 2013: 36–51 and ch. 6). The aesthetic premise
presents a very different problem. Consider Wittkower’s statement: ‘I
think it is not going too far to regard commensurability of measure as the nodal
point of Renaissance aesthetics’ (1953:
16). One cannot logically propose a causal relationship between the
dimensional properties of a proportional system and an assessment of architectural
aesthetics, any more than one can attribute dimensional qualities to an idea, except
metaphorically. Yet Wittkower is not writing metaphorically here — he is
conveying his genuine belief that a causal relationship exists between the
quantitative and the qualitative; between numbers and opinions about architectural
beauty. It is a metaphysical belief that, when observed in
mid-20th-century and later contexts, I call ‘proportional aesthetic
mysticism’. As a modern phenomenon it thrives in part due to the high-volume,
Wittkower-influenced textbooks that continually shape the thinking of new
generations of art and architectural historians worldwide (Cohen 2013: 17–18, 281–287). The persistence of
this type of mysticism owes more, however, to its deep roots in Western culture. In
proportional aesthetic mysticism the association of numbers with beauty exploits a
cognitive ambiguity reflected in the very languages that help millions of people
form their thoughts. When speakers of at least English and the Romance languages say
the word ‘proportion’, two contradictory meanings fire off
simultaneously in their minds, and seem to meld together: the concepts of ratio and
beauty become unified.

Two kinds of proportion

Proportion technically denotes a ratio, but in common usage it usually connotes a
broader meaning that appears to have entered the English language, with all its
current ambiguity, with Ephraim Chambers’s 1723 translation of the French
Traité d’architecture of 1714 by Sébastien Le
Clerc: ‘By Proportion I don’t here mean a Relation of Ratios as the
Geometricians do; but a Suitableness of parts, founded on the good Taste of the
Architect’ (Le Clerc, 1723–1724, 1:
29).30 The term thus embodies what I
call ‘quantitative/qualitative ambiguity’ because the first meaning,
which I call ‘proportion-as-ratio’, is an abstract quantitative
comparison, while the second, which I call ‘proportion-as beauty’, is a
qualitative aesthetic assessment of an identified object.31 The former cannot bring about the latter, any more than the
latter can result from the former, because objective numerical relationships cannot
cause, predictably and repeatedly, specific subjective
emotional responses such as the opinion that a building is beautiful. When the
beauty-in-proportion belief system originated centuries ago, however, the notion of
beauty was not always considered to be a subjective product of human emotion.

Antecedent to Le Clerc’s dualistic description of proportion is Claude
Perrault’s similar observation, in the preface to his
Ordonnance of 1683, that ‘there are two kinds of
proportion’. The first, which Perrault notes is difficult to perceive,
describes ‘the magnitudes that the various parts [of a building] have in
relation to each other or to the whole’, while the second, which he notes is
called Symmetrie [sic] in French, consists of ‘the
relationship of all the parts together [… and] is a very apparent thing, [for
…] it never fails to make apparent the defects’ in a building (Perrault 1993: 50–51).32 Thus, according to Perrault, the first kind of proportion
consists of a series of quantitative relationships that are difficult to see with
the unaided eye and presumably can only be revealed by measuring instruments, while
the second is a relationship among the parts of a whole that is universally
distinguishable by all human beings as either aesthetically correct or defective. We
need only observe, however, that as an example of one of the defects that the second
kind of proportion purportedly reveals, Perrault describes the interior of the
Pantheon — ‘the bands of the dome do not correspond with the windows
below, causing disproportion and a lack of symmetrie that everyone
can readily recognize’ — to confirm that aesthetic judgments are ever
subjective from time to time, place to place, and individual to individual.33 Indeed, just over three centuries later,
Howard Burns, in his university classes, often singled out this very misalignment as
one of the aesthetically successful features of the Pantheon,
because, he contended, it makes the dome appear detached from the cylinder and
freely rotating, as if floating (Fig. 4).34

Fig. 4

The Pantheon, Rome, interior view. Photo: Emilio Labrador.

Perrault’s inability to understand the second of his two kinds of proportion as
a subjective aesthetic assessment, rather than as a universal ‘that everyone
can readily recognize’, was his blind spot, and helps to explain why the
distinction he also made between two kinds of beauty, arbitrary (i.e., learned) and
positive (i.e., universal), according to which he controversially associated
architectural proportional systems with the former, created no significant
impediment to the continuation of the beauty-in-proportion belief system into the
present day. His theory, though radical in his day, did not attack the core of the
beauty-in-proportion belief system. Indeed, it was never his intention to undermine
the very notion of positive beauty, but only the belief that proportional systems
could be sources of positive beauty in architecture.35

Perrault’s notion of positive beauty — i.e., beauty as an objective
entity not unlike a mathematical principle, unreliant on subjective human judgment
for its existence but universally recognizable as beauty by all human beings —
happens to be the necessary precondition for the belief that particular proportional
systems create beauty in architecture, or, the very belief that he developed the
notion of arbitrary beauty to combat.36
Perrault could very well claim that the proportional systems of the past, and for
that matter his own new proportional system for the orders presented in the main
body of the Ordonnance, only create beauty arbitrarily, through the
familiarity of custom; but his affirmation of the existence of positive beauty has
only affirmed the core belief of beauty-in-proportion believers from his day to our
own: that a metaphysical well of ideal beauty exists somewhere outside of
architecture, and that architects can learn various ways in which to tap into it in
order to create works of universal appeal. Perrault attacks the efficacy of one of
those ways — the use of proportional systems — but not the core of the
belief itself.

Perrault’s notion of positive beauty is similar to Leon Battista
Alberti’s notion of innate beauty, and might strike the modern reader as a
contradiction in terms, for beauty, it would seem, can never be positive.37 All assessments of beauty are arbitrary
aesthetic opinions — at least, this contention could not be disproven based on
logical argument or scientific demonstration. Perrault’s conception of
positive beauty is based on a blending of quantitative and qualitative architectural
qualities, or ‘convincing reasons’, that to him signal the presence of
positive beauty. He provides four examples of such qualities: ‘richness of
materials, the size and magnificence of the building, the precision
and cleanness of the execution, and symmetrie’. The first and
third of these examples can be interpreted either as measurable qualities or
subjective judgments. The second combines a measurable quality, size, with a
subjective judgment, magnificence, and is thus best interpreted as
a pair of words referring to the subjective quality of
magnificence. Perrault describes his fourth example,
symmetrie, as the same as his second kind of proportion, or, a
quality that produces ‘an unmistakable and striking beauty’ (‘une
beauté evidente & remarquable’) that all people recognize.38 Thus Perrault’s positive beauty would
seem to consist of but a series of arbitrary aesthetic assessments.

Perrault did not consider qualities such as magnificence and
symmetrie to be subjective aesthetic judgments, even though
today we have no other way to characterize them because their properties cannot be
confirmed with the predictability and repeatability that the scientific method
requires. Indeed, he based his assumptions not on scientific standards of
verifiability, but on the then seemingly irrefutable approbation of expert opinion
— the widespread consensus among those who had the education and training to
judge art and architecture. Another century would pass before Kant would state
that

there is no Science of the Beautiful, but only a Critique of it […]. For
[…] if it could be decided scientifically, i.e. by
proofs, whether a thing was to be regarded as beautiful or not, the judgment
upon beauty would belong to science and would not be a judgment of taste. (Kant 1914: 185)39

Perrault did not have the benefit of the fully mature Scientific Revolution to help
him sort out these distinctions, but no matter, because the concept of
beauty-in-proportion, which depends on the illogical and unscientific assumption of
a causal relationship between proportion-as-ratio and proportion-as-beauty, ignored
the Scientific Revolution in its uninterrupted passage from Perrault’s day to
our own.40

By codifying the notion of positive beauty, and thus the positive/arbitrary beauty
dichotomy, Perrault’s writings may have contributed to maintaining the
beauty-in-proportion belief system in subsequent centuries as much as those of
François Blondel.41 In Part V of the
Cours d’architecture of 1683, Blondel replies to
Perrault’s denial, in Perrault’s preface to the
Ordonnance, that proportional systems can be sources of beauty
with what Anthony Gerbino calls a ‘defense of proportion’.42 Blondel’s defense focuses on
‘harmony’ (harmonie), an adjunct to the word and
concept of proportion that for architectural theorists had carried the ambiguous
double signification of proportion-as-ratio and proportion-as-beauty since at least
1485, when Alberti published his celebrated promulgation of harmonic architectural
proportions in Book IX of De re aedificatoria.43 Thus in one sentence Blondel uses both of these terms,
proportion and harmony, first qualitatively, to describe the beauty of ‘old
and modern buildings […], [and] the beautiful proportions that their parts
have between them […] which have [… an] agreeable harmony that gives so
much pleasure to the eyes’; and, in another sentence on the following page,
quantitatively, in reference to the beautifying qualities of musical-numerical
proportions of a specific building as ‘a continual harmonic proportion’
(Blondel, 1675–1683, vol. V,
738–739).44

Blondel emphasizes his belief that an inherent beauty of harmonic ratios in music is
directly transferable to architecture in an unsubtle graphic comparison between the
horizontal lines of a column base, annotated with numerical dimensions that form
harmonic ratios, with the lines of a musical staff. The staff poignantly includes a
bass clef (Fig. 5; Blondel, 1675–1683, vol. V, 759).

Blondel’s claim that musical harmonies contain inherent beauty that can be
transferred to architecture is not fundamentally different than Perrault’s
claim that magnificence and symmetrie, for
example, serve as vehicles for transferring positive beauty to architecture, for
both authors believe that great works of architecture somehow access a metaphysical
well of ideal, universal beauty. Perrault’s skeptical approach toward the
traditional notion of proportional systems as tools capable of tapping into that
well did not extend to the notion that such a well, which he calls positive beauty,
existed in the first place. Thus, in the aftermath of the highly visible
Perrault-Blondel debate, Perrault’s skeptical approach to the traditional
association of proportional systems with positive beauty did not win out over
Blondel’s respectful approach. Perhaps it was not different enough from
Blondel’s approach, or perhaps the beauty-in-proportion belief system, then as
now, was impervious to any logical counter argument, including those parts of
Perrault’s counter argument that are indeed extremely logical.45 Instead, these two general approaches —
skeptical and respectful — continued into subsequent centuries as parallel,
sometimes mutually antagonistic developments of architectural culture.

Wittkower surveys the European literature that reflects these parallel developments
from the 17th through the early 20th centuries — though rather than concurrent
developments of two different approaches Wittkower sees a transition from one to the
other — in a section of Architectural Principles titled
‘The Break-away from the Laws of Harmonic Proportion in Architecture’
(1949 and 1952b: 124–135; 1962 and
1971: 142–154). The literature he
surveys is wonderfully varied, ranging from the analytical to the whimsical, and
most of it highlights the notion of harmony.46
An important shortcoming of Wittkower’s overall quite useful survey, however,
is his conclusion that beginning in England in the late 18th century, ‘the
whole structure of classical æsthetics was overthrown from the bottom’,
and that

in this process man’s vision underwent a decisive change. Proportion became
a matter of individual sensibility and in this respect the architect acquired
complete freedom from the bondage of mathematical ratios. [Footnote 4:] However,
mathematical ratios survived in a degenerated form as a teaching expedient for
architectural students and without any connection with their original meaning.
(Wittkower 1949 and 1952b: 131 and 134; 1962 and 1971: 150 and
153)

Wittkower’s freedom/bondage dichotomy appears to be overstated, for pre-18th
century architects appear to have had more proportional freedom than Wittkower
acknowledges, and while later architects may indeed have acquired the option of
freedom from belief-based proportional systems (see below), not all of them opted
for it; and of course architects were not the only interested parties in the history
of architectural proportional systems. For some 18th-century and later thinkers,
proportional systems continued to carry the same general payload of metaphysical
meanings that they had carried for some thinkers of preceding centuries. Thus it may
be more useful to think of the history of architectural proportional systems as
characterized by two continuous, parallel strands of thought — a
skeptical-pragmatic strand and a respectful-metaphysical strand — rather than
a transition from one way of thinking to another, characterized by an 18th-century
sea change separating a long period of universal obedience to proportional system
metaphysics from a modern period of liberation.

Wittkower’s denial of pluralism in European attitudes toward architectural
proportional systems during the centuries in question is reflected in his survey
selections. The earlier works included in Wittkower’s survey tend to reflect
the respectful-metaphysical strand, while his later selections mostly reflect the
skeptical-pragmatic strand. Wittkower brings his survey only as far as Archibald
Alison’s Essays on the Nature and Principles of Taste of 1790
and his follower Richard Payne Knight’s An Analytical Inquiry into the
Principles of Taste of 1805, before proclaiming victory for the
skeptics. He then provides proportion-skeptical quotations from two subsequent
works, John Ruskin’s The Seven Lamps of Architecture of 1849,
and Julien Guadet’s Eléments et théorie de
l’architecture of 1901–1904, as examples, he claims, of the
‘general feeling’ prevailing from Knight’s day ‘down to our
own days’ (Wittkower 1949 and 1952b: 134; 1962 and 1971: 154). He neglects
to acknowledge, however, that many other works from the 19th and early 20th
centuries reflect a broad range of vigorous alternative views.47

Perhaps fueled by an enduring tradition of occultism, which had flourished in England
with particular fervor during the late 17th and 18th centuries, English
beauty-in-proportion believers in the mid-19th century appear to have been very
active, the legacies of the influential, proportion-skeptical writings of William
Hogarth (1753), David Hume (1739 and 1757), and Edmund Burke (1757) cited by
Wittkower notwithstanding.48 Thus Edward Lacy
Garbett, in his Rudimentary Treatise on the Principles of Design in
Architecture, published in London in 1867, decries the ‘immense
abuse’ he attributes to the beauty-in-proportion believers of his day, in a
passage that he subsequently supports with quotations from Alison’s
aforementioned Essays. Garbett writes,

A proper understanding of the nature of physical harmony, whether in sound or
colours, will guard the reader against the immense abuse which mystics make of
this plain commonsense principle, in the theories of what is called
proportion in architecture; — a sort of beauty made
easy, an artistic philosopher’s stone, by which baser productions are to
be transmuted into works of art […] only by applying arithmetical rules.
(Garbett 1867: 38–39)

John Pennethorne’s impressive work, The Geometry and Optics of Ancient
Architecture of 1878, indicates that Garbett’s mystics were not
relegated to the fringes of English society. Pennethorne pairs his archaeological
observations, which are still important today, and which he presents in rigorous,
large format measured drawings and lucid verbal descriptions, with extensive
metaphysical reflection. His first mention of optical corrections in ancient Greek
temples as being necessary ‘to produce an apparent harmony between all the
members of the executed design’ seems to refer to harmony as visual beauty in
a casual, non-metaphysical way (Pennethorne 1878:
4). He continues, however, in the universalizing first person plural
(‘we’), to claim that rather than perceiving harmony in universally
appreciated works of art merely with our eyes, we feel it through
an occult sympathy with the ‘constitution of our minds’, which contains
‘an original impression’ of the inherent structure of the universe. Thus
he claims, as part of a lengthy metaphysical declaration strongly reminiscent of
Morris’s Lectures from 150 years earlier, that when we (i.e.,
all human beings)

are able to perceive the harmony and the exact proportions in which the several
parts of the Universe are linked together, we feel an intellectual pleasure,
arising perhaps from an original impression on our minds of what appear to be
the essential attributes of a perfect work. (Pennethorne 1878: 45)49

Another expression of belief in the intangible, beneficial properties of proportional
systems evident in mainstream, 19th-century English architectural theory will be
discussed below, in relation to Gwilt’s Encyclopedia.

The ambiguous, metaphysically driven melding of the two kinds of proportion discussed
in this section, proportion-as-ratio and proportion-as beauty, have found four main
categories of expression in the art and architectural literature from Alberti to the
present. Two of them, we have seen, are the terms ‘proportion’ and
‘harmony’.50 Indeed, today
scholars and architects still commonly refer to ‘harmony and proportion’
without understanding specifically what these words mean, or realizing that by using
them they are perpetuating an ambiguity that traces back at least as far as the
early Renaissance. The third category is the notion of regulating lines, which along
with harmony Blondel also promotes in his Cours (Fig. 6; Blondel, Cours, V.ix.752).
The fourth, the virtual cult of the golden section, originated in Germany in the
mid-19th century, and is only superficially related to the occasional and probably
often inadvertent appearance of this ratio (1:1.618…) in medieval
architecture.51

These four categories of expression blended with particular fervor in France during
the second decade of the 20th century, in the discussions of several avant-garde
groups composed of artists and others, including Section d’Or (Golden Section;
also called Groupe de Puteaux), Les artistes de Passy, and Art et Liberté.
Among the various members and officers of these groups were August Perret, Paul
Valéry, Amédée Ozenfant, Gino Severini, Pablo Picasso and
Charles-Édouard Jeanneret-Gris, the future Le Corbusier (Laurent 1998; Loach
1998; Jeunes Peintres ne vous frappez pas!
1912).52 Out of this early
20th-century French cultural context, augmented by German and other influences,
eventually emerged Le Corbusier’s Modulor, a proportional system of the 1940s
that combined all four of the above-noted categories of expression of the
ambiguously quantitative/qualitative notion of proportion.53 Out of this context also belatedly emerged, in 1951, the
Milan conference, which for some of the older participants such as Severini and Le
Corbusier must have carried a rather nostalgic air of reunion, albeit in a
dramatically different, post-war world (Fig. 2).54

The legacy of the Milan conference

Dominated by the conspicuous participation of Le Corbusier and Wittkower, and given
augmented prestige by the contributions of other leading intellectuals including
Sigfried Giedion, Matila Ghyka, Pier Luigi Nervi, Andreas Speiser and Bruno Zevi,
the 1951 conference gave voice to a spiritual yearning on the parts of the
organizers and participants for the development of a unified, orderly basis for the
arts and sciences as a pathway toward the reformation of society, and ultimately,
recovery from the trauma of World War II.55 In
his introduction to the recently published proceedings of the conference, Fulvio
Irace describes this yearning as follows:

In 1951 the conference De Divina Proportione was proposed as an
ecumenical council of men of arts and sciences, convened to determine the rules
of the spirit that were to govern the new areas of the reconstruction of
democracy. (Cimoli and Irace 2007:
17)56

In the same publication James S. Ackerman — today the only living contributor
to the 1951 conference and also a contributor, via video interview, to the 2011
Leiden conference — similarly notes a spiritual dimension to the
conference:

The interest that arose in 1951 was perhaps born, in a Europe that was still
searching to recover from the devastation of the war, from a desire to return
spirituality to the arts and to life through the geometry of a pure
architecture, free of ornament and consisting of rectangular surfaces and
openings. (Cimoli and Irace 2007:
34)57

Ackerman later notes that the manner in which the conference participants approached
their subject, if not the idealism that motivated them, marked the beginning of a
new scholarly seriousness in the study of proportion:

Before that time [1951] it [proportion] really hadn’t become a reliable
[area of] study. There was a lot of mysticism around it. Some of the mystics
were part of the conference too, which is only fair, but it was really the end
of the mystical phase and the [beginning of the] effort to set it onto reliable,
academic, practical grounds.58

Only tangentially, however, did the conference engage the academic study of the
history of proportional systems. Those contributors who incorporated historical
observations with supportive textual references into their presentations, in
particular Wittkower and Giedion, only did so in support of the overwhelmingly
mystical, reformist agenda of the conference, which Giedion rather grandiosely
described as ‘revolutionary’ (‘Il tutto e la parte
nell’architettura contemporanea’, in Cimoli and Irace 2007: 75). Wittkower, one of the conference organizers,
justified that agenda in his opening remarks by decrying as ‘an
illusion’ what he saw as the predominant contemporary attitude toward artistic
production based on ‘the nineteenth century idea that the artist, in his
creative act, should be guided only by his personal intuition’. On the
contrary, he declared, ‘the search for harmony and order is a basic part of
human nature’ (‘Finalità del Convegno’, in Cimoli and Irace 2007: 47). Such harmony and
order, he believed, transcended the individual, and had the potential to be
perceived collectively, by all human beings.

Indeed, for Wittkower and the other conference participants, a general notion of
proportional order, which could be manifested in proportional systems and which they
called the divina proportione (the divine proportion), after Luca
Pacioli’s 16th-century book of that title (1509), constituted a kind of
demiurge, existing independent of human culture but occasionally interacting with
it. Seemingly endowed with agency and thus more assertive than a passive set of
Platonic ideals, the divina proportione, these participants
believed, periodically appeared in history, demanding expression in the arts and
compelling human beings to serve as its sometimes unwitting collaborators toward
some mysterious but ultimately beneficent purpose. Thus Giedion provocatively asked
in his conference paper, ‘Can we state that the divina
proportione has made its appearance again?,’ and obliquely
answered in the affirmative, citing as examples Le Corbusier’s Modulor and
‘the difference between the static proportions of the past and the dynamic
proportions of the present epoch’ (Giedion, in Cimoli and Irace 2007: 73–74).59 Even the thirty-two-year-old Californian and recently discharged US
Army enlistee James S. Ackerman got into the spirit of the conference, alongside his
elder European colleagues, concluding his summary analysis of the Cathedral of Milan
proportions by interpreting the various geometrical schemes documented in the
cathedral archives as medieval expressions of ‘the “Divina
Proportione”’ (Ackerman, ‘Le
proporzioni nell’architettura gotica: Milano, 1400’, in Cimoli and
Irace 2007: 51).60

Wittkower shaped the conference around the goal of identifying an appropriate
expression of the divina proportione in the arts for the modern
age. Since ‘the artist reflects the culture in which he lives’, he
posited, the central objectives of the conference were, first, to answer the
question ‘What is the character of our culture?’ in light of ‘the
substitution of the absolute measure of space and time with the new dynamic
space-time relationship’ introduced by Einstein; and second, to determine what
effect this substitution ‘has and will have on proportion in the arts’
(Wittkower, ‘Finalità del Convegno’, in Cimoli and Irace 2007: 47). Thus, at the Milan conference
Wittkower played the role of the activist-historian, applying his historical
knowledge toward the purpose of influencing rather than merely studying history.
Through the conference he strove to encourage artists and architects of the time to
develop new proportional systems that would reflect the contemporary modern
condition, to use those proportional systems in their creative works, and to see
themselves as the torch bearers of a dynamic, centuries-long tradition of
proportional exploration that had been, in his view, temporarily interrupted by
misguided 19th-century attitudes toward creative production.61

In his 1960 essay ‘The Changing Concept of Proportion’, Wittkower reveals
his disappointment with the 1951 conference, lamenting that it had failed to advance
its reformist agenda with tangible results. He also reveals his belief that
proportional systems constituted not merely opportunities for aesthetic expression,
but moral imperatives. The Milan conference, he notes, ‘brought together
philosophers, painters, architects, musical historians, art historians, engineers
and critics from many countries’. These thinkers and practitioners had
gathered because, he continues, ‘they agreed on one point: that some kind of
controlling or regulative system of proportion was desirable’. The conference
nevertheless ‘fizzled out’, he claims, ‘without making an
appreciable impact on the younger generation’ (Wittkower 1960: 210). The true depth of his disappointment, however,
becomes apparent as his essay continues.

‘The bankruptcy of the Milan meeting’, Wittkower inveighs, ‘was
publicly sealed at a historic meeting of the Royal Institute of British Architects
[…] where a debate took place on the motion “that systems of proportion
make good design easier and bad design more difficult” — a motion that
was defeated with forty-eight voting for and sixty voting against’ (Wittkower 1960: 210; Pevsner 1957a).62 This
‘bankruptcy’ was for Wittkower not merely intellectual, but moral. In
the essay he appeals for a return to the high ideals of the failed conference,
advocating adherence in contemporary art and design to ‘absolute’ and
‘universal’ values based in ‘thought’ rather than
‘sensations’, lest modern society succumb to ignoble pragmatism and
opportunism. Wittkower continues:

In most periods of history artists were convinced that their specific system of
proportion had universal validity. These systems derived their all-embracing
character from thought processes rather than from sensations. It is now two
hundred years since the belief in absolute values was shaken, perhaps for all
time; it can surely not be won back by an act of majority decision. As long as a
broad foundation for a resurrection of universal values is lacking, one cannot
easily predict how the present dilemma can be resolved. The very formulation of
the motion put before the R.I.B.A. meeting shows that we have left far behind
the realm of the absolute, and are submitting to pragmatic and opportunistic
motivations. (Wittkower 1960: 210)

With the emphasis of military and religious metaphors (‘won back’ and
‘resurrection’, above), Wittkower here presents a moral choice between
good and bad: design with proportional systems is based on thought and thus reason,
and is therefore good; design without proportional systems is based on aesthetic
judgments that are in turn based only on stimuli received by the senses in the
absence of thought, and is therefore bad. He goes on to lament the ‘quick rise
and easy victory of abstract expressionism’, which he deprecates as
‘splash-and-dribble style’, and the ‘absolute subjectivism’
that he felt characterized the state of society nearly a decade after the Milan
conference, and that he considered antithetical to the use of proportional systems
(Wittkower 1960: 210).

With these comments Wittkower takes his place in a long line of like-minded thinkers.
He might have fit in comfortably, for example, with those who, in 1750, Alexander
Gottlieb Baumgarten anticipated might raise objections to his proposed new field of
aesthetics on the basis that ‘impressions received from the senses, fantasies,
emotional disturbances, etc., are unworthy of philosophers and beneath the scope of
their consideration’ (Baumgarten, ‘Prolegomena’ to his
Aesthetica, translated and quoted in Harrison, Wood and Gaiger 2000: 490).63 In earlier centuries Wittkower might have found sympathetic
company with François Blondel or Daniele Barbaro. This tendency to think of
proportional systems, and the buildings that contain them, as good because they are
based on mathematics, and the absence of proportional systems as less good, if not
outright bad, because it leaves the architect’s whims unfettered, is still
common today. Indeed, this tendency, along with the proportional aesthetic mysticism
of which it is a symptom, carries the risk of encouraging moral aesthetic judgments
of architecture, along the lines of Wittkower’s above-quoted comments of 1960
pertaining to the Milan conference. If buildings that contain proportional systems
are good according to beauty-in-proportion believers, can buildings that lack them
ever be as good, of equal overall value, and more than merely
‘pragmatic’ and ‘opportunistic’? Such proportional aesthetic
mysticism could lead some scholars to believe that all good buildings must
necessarily have interesting proportional systems, and to insist on finding them
even where they do not exist. As my study of the Old Sacristy of San Lorenzo
suggests, however, many good buildings may very well lack interesting proportional
systems, and that lack constitutes valuable historical information rather than
grounds for censure (Cohen 2013:
140–145).

Belief-based proportional systems

Prior to the advent of modern structural engineering, architects and builders used
proportional systems to determine key dimensions of their works in terms of local
units of measure. They did so in the belief, which could never be based on the
certainty of verifiable outcomes, that proportional systems would confer upon their
works certain desirable but unmeasurable qualities.64 Such qualities included a general condition of order that was
integral to pre-engineering notions of structural stability, beauty and overall
correctness, which in Italy was called ordine.65 After the advent of engineering new kinds of proportional
systems arose based on the measurable, scientifically verifiable certainty of
guaranteed outcomes, while examples of the old kinds based on uncertain,
unscientific beliefs continued to flourish, though to a lesser extent, alongside
them. In acknowledgement of this variety, we may define an architectural
proportional system as

a set of geometrical, numerical or arithmetical correspondences between important
dimensions throughout a building or major part thereof, intended by the
architect to imbue built form with desirable qualities, physical or
otherwise.66

Note that in order to satisfy this definition, either kind of proportional system
must consist of a set of intentional correspondences. This
definition thus excludes complex geometrical constructions that historians might
overlay onto drawings, photographs or computer models of buildings, unless those
overlays can be convincingly demonstrated, through building measurements combined
with other evidence, to represent the architect’s intentions.67 This definition thus furthermore assumes that
proportional systems cannot wander into architecture of their own volition, without
the architects’ knowledge, as for example golden sectionists have tended to
believe.68 Since unintentional patterns of
geometry and number can always be found in architecture, the preceding definition
distinguishes between mere physical description of the object, which might include
such patterns, and the scholarly identification and analysis of the creative
intentions of the architect.69

The use of architectural proportional systems continues to be standard practice
today, in the forms of structural engineering size specifications (which must be
combined with specifications for materials, techniques and other construction
factors), standardized sets of dimensions for building components (in terms of the
meter or the foot), zoning regulations (such as floor area ratio) and other
conventions, but these are not the kinds of proportional systems of primary interest
to us here.70 Engineering specifications,
dimensional standardization and zoning formulae are designed to guarantee
measurable, practical outcomes, repeatedly and predictably. The proportional systems
of primary interest to us here are decidedly impractical and not founded on the
certainty of guaranteed outcomes.71 These
proportional systems either date to the pre-engineering period prior to
1742–1743, or if later, retrogressively retain the technological innocence of
that earlier period, together with some degree of the metaphysical orientation that
characterized much of the thinking of that long period. I will call them
‘belief-based’, as opposed to ‘certainty-based’,
proportional systems.72

The differences between belief-based and certainty-based proportional systems can be
rather subtle. For example, upon first consideration one might think that the San
Lorenzo nave arcade bay proportional system (Fig. 3), which is a pre-engineering proportional system, guarantees the
outcome that a root-2 rectangle will be inscribed in the space between the column
shafts, and that therefore at least in this respect it is no different than any
typical engineer’s specification from the post-1742–1743 period. Upon
closer consideration, however, we can see that this proportional system is
thoroughly belief based rather than certainty based, for two reasons. First, it does
not in fact guarantee the physical presence of the root-2 rectangle in the nave
arcade bays — indeed, this rectangle is not physically present there at all
— and second, this root-2 rectangle could serve no practical purposes whether
it were present or not. This rectangle only exists as an idea in
the minds of observers who understand that the horizontal distance between the
column plinths, and the intended height of the column shafts, not including the
astragals (which in this basilica are physically integral with the capitals),
together correspond to the proportions of an imagined root-2
rectangle. To mentally perceive this rectangle observers have to understand that
they must disregard not only the notable gaps between the column shaft surfaces and
the sides of this imaginary rectangle (Fig. 3),
but also the construction error that caused the column shafts to have been made
slightly too tall to mark the top of this imaginary rectangle (Cohen 2013: 104–111; 2008: 33–37).

Furthermore, the root-2 rectangle in question could not have
guaranteed any of the outcomes that the architects, probably
both Dolfini and Brunelleschi, might have hoped to achieve by specifying it, which
was surely more than simply creating a root-2 rectangle as an end in itself.73 These architects could very likely have
intended, for example, that it would confer ordine, specifically
including structural stability (see above). Today, however, we know that it is
scientifically impossible for a root-2 rectangle per se to
establish structural stability in architecture; and neither ordine
nor other notions of beauty, being subjective qualities, can ever be guaranteed.
Engineering or zoning specifications, conversely, being based on immutable
scientific and municipal laws (the latter being immutable at least for the duration
of construction), guarantee predictable and measurable outcomes such as structural
stability (when used in conjunction with other specifications, as noted above) or
conformance with established building codes.

An illuminating example of an early conflict between belief-based and certainty-based
proportional systems is found in the 1867 edition of Gwilt’s
Encyclopedia. Gwilt promotes what he calls the ‘interaxal
system’, a grid proportional system that he acknowledges having borrowed from
Durand’s Preçis des leçons d’architecture of
1802–1805. Durand’s proportional system still represented the
belief-based approach, and Gwilt defended it against the then-new engineering
technology of cast iron, which threatened to deprive Durand’s system of the
structural justification with which Gwilt associated it (Durand (1802–1805).74 Gwilt writes,

Not the least important of the advantages resulting from the method of designing
just submitted to the reader is the certain symmetry it produces, and the
prevention, by the use of these interaxal lines on each floor, of the architect
falling into the error of false bearings, than which a greater or more dangerous
fault cannot be committed, more especially in public buildings. The subterfuge
for avoiding the consequence of false bearings is now a resort to cast iron, a
material beneficially enough employed in buildings of inferior rank; but in
those of the first class, wherein every part should have a proper point of
support, it is a practice not to be tolerated. (Gwilt 1867: 894–895)

Thus, according to Gwilt, not only does the interaxal system provide a ‘certain
symmetry’, by which Gwilt evidently means a kind of comprehensive beauty
similar to Perrault’s symmetrie, but it ensures that walls
and columns will always be stacked directly atop other walls and columns, in a
system of structural support that Gwilt finds more satisfactory than one that uses
columns and transfer beams of cast iron, even though the two systems could be made
equally strong. Similar to Wittkower’s above-quoted objections of 1960 to any
neglect of belief-based proportional systems in favor of artistic intuition, Gwilt
considers the replacement of a belief-based proportional system with an
engineer’s certainty-based one (i.e., the mathematical specifications for the
cast iron members) to be morally unacceptable. The interaxal system, Gwilt says,
provides structural support that is ‘proper’, while cast iron provides
the same degree of support but only though ‘subterfuge’. For Gwilt, the
engineer’s cold calculations, which merely satisfy the practical objective of
making a building stand up, can never distinguish architecture — i.e.,
buildings of ‘the first class’ — from mere ‘buildings of
inferior rank’, or for that matter, from pure works of engineering such as
bridges.75 Gwilt’s protestations
notwithstanding, 19th-century engineers indeed succeeded in robbing belief-based
proportional systems of their one ostensibly practical purpose, that of ensuring
structural stability, by fulfilling that purpose effectively and reliably using
proportional systems based on the science of physics, which the old proportional
systems based on the mysticism of metaphysics never could do.

The advent of engineering, however, brought about only a partial demise of
belief-based proportional systems in architecture. Since such systems, like those
used in the designs of the Cathedral of Milan and the basilica of San Lorenzo, never
had any more influence over structural stability than prayer or luck, the advent of
engineering merely proved what many architects already knew — that in
determining the sizes of crucial structural members an architect could use all the
proportional systems he wanted, but in the end, in the words of the 16th-century
Spanish architect Rodrigo Gil de Hontañon, he could only use ‘his own
judgment […] and dare to have confidence’ (quoted in Kubler 1944: 146).76 The advent of engineering thus robbed belief-based proportional
systems of their always-questionable claims to have helped ensure structural
stability, but did not touch the only purpose that such proportional systems have
ever fulfilled successfully — that of imbuing buildings with meaning. Some of
that meaning may be considered aesthetic, for example, when observers thought about
proportional systems in order to help themselves make sense of sense perception (Van
Eck, ‘The Composto Ordinato’ in this special
collection), or when architects used proportional systems to establish certain forms
like entasis that they considered to be beautiful; and some may be considered
metaphysical but not necessarily aesthetic, such as when observers thought about
proportional systems to help themselves imagine architecture as a reflection of a
larger, macrocosmic order that, in the words of Alfred W. Crosby (1997: 46–47), ‘lay beyond the scrim
of reality’.

Belief vs. practice

Contemplation of the macrocosmic order that lay beyond the scrim of reality was not
for everyone, and most pre-engineering architects probably used belief-based
proportional systems without associating them with beliefs that were nearly as
metaphysical as the kind Crosby describes above, who addresses a more general
context not specifically focused on proportional systems. Indeed, Anthony Gerbino
and Konrad Ottenheym both independently conclude that in the 17th century and
earlier periods architects probably had little interest in, nor much understanding
of, such beliefs, and instead thought of proportional systems as practical design
tools integral to long-established architectural practices.77 Palladio and Vignola, for example, in their extremely
limited comments on possible analogies between proportional systems and musical
harmony, merely indicate a general awareness of such matters, and that others of
their day had studied them, but devote the vast majority of their own attention to
more earthly concerns of architectural practice.78 In Vignola’s case one of those concerns is his attempt to use
proportional systems to establish architectural beauty through
correlation — that is, by recording the proportions of
selected ancient Roman buildings widely considered beautiful in his day, and
encouraging his contemporaries to use those proportions in their own works (Da Vignola 1562: Prefazione, n.p.).
Vignola’s belief in the power of proportional systems to contribute to
architectural beauty is illogical, for it fails to acknowledge the myriad factors
that together contribute to perceptions of architectural beauty, perhaps including
certain acceptable ranges of proportions-as-ratio for particular architectural
elements as determined by custom.79 His belief
is not fully metaphysical in character, however, unlike Alberti’s stated
belief in the powers of proportional systems to create architectural beauty through
causation, or, through the sheer metaphysical power of
numbers.80

Thus as an alternative to Wittkower’s monolithic interpretation of
architectural theory, according to which virtually everyone in any given period
thought in exactly the same way, and according to which the extreme views of
non-architect mystics such as Francesco Giorgi can serve as reliable stand-ins for
the views of pragmatically inclined architects such as Palladio, Vignola, Serlio and
Alberti (considering De re aedificatoria, Books I to XIII and X), I
have proposed above that at any given time in history two rather loosely defined,
parallel strands of belief pertaining to proportional systems can be identified, one
more pragmatic in character and the other more metaphysical (Wittkower 1949 and 1952b:
90–102, 136­–138; 1962 and 1971: 102–116,
155–157). In the pragmatic strand are the beliefs that proportional
systems contributed various degrees of structural stability, beauty and
ordine to architecture — beliefs held, for example, by
the aforementioned three Renaissance architects, plus Alberti, depending on how we
think about him.81 In the more metaphysical
strain are the beliefs that proportional systems link architecture to the macrocosm
or the divine, such as, continuing with Renaissance examples, those of Barbaro,
Giorgi and Alberti, considering De re aedificatoria, Book IX.
Indeed, the beliefs of most pre-engineering architects and builders can probably be
identified with the pragmatic strand, while those of unusually learned architects,
clerics and other intellectuals, with the metaphysical strand. As noted above,
furthermore, some extraordinary thinker-practitioners such as Alberti and Le
Corbusier can be interpreted as occupying both strands, depending on which aspects
of their work we consider.

Of course, the character of the various beliefs within each strand can be expected to
have varied considerably across time and geography. Thus Pennethorne’s
beliefs, for example, were no doubt quite different from Giorgi’s, though both
contributed to the metaphysical strand. Nevertheless, Pennethorne’s may be
considered more closely related, through a continuous succession of metaphysically
oriented thinkers from the 19th back to the 16th centuries, to Giorgi’s
beliefs than to Palladio’s, whose more pragmatic orientation associates him
with the pragmatic strand that also stretches from the Renaissance into the 19th
century, but was manifested by more pragmatically oriented thinkers such as Alison
and Knight.82

And so the parallel strands of belief continued, progressing out of the 19th century
and into the 20th. In September 1951 the metaphysical strand passed forcefully
through Milan, though the pragmatic strand was also present.83 In June 1957 both strands passed through the RIBA meeting
moderated by Pevsner, though we have seen that the pragmatic strand appears to have
been slightly more vigorous, at least in light of the 60–48 vote against the
beauty-in-proportion belief. When interpreting strands of belief we must exercise
due caution, for there is reason to question the depth of beliefs to be found among
those thinkers we may associate with the metaphysical strand. The British
architectural journals from the late 1950s, after all, despite the sizable minority
of the RIBA vote, are not flooded with articles about proportional systems in
practice. In architecture the demands of practice have always tended to hold
esoteric beliefs in check, which is why most practitioners in history have tended to
associate with the pragmatic strand.

Today, while a majority of architects seem to believe that proportional systems
contribute beauty to architecture of the pre-engineering period, very few of them
use belief-based proportional systems, at least of the old-fashioned varieties such
as those involving regulating lines, harmonic numbers or the golden section, in
their own work as supplements to the certainty-based proportional systems that
everyone uses.84 To believe that the Parthenon
is beautiful due to some secret proportional systems of the ancient Greeks may be
entertaining and satisfying enough, but for an architect today to attempt to use
such proportional systems in his or her own work would not only require more
specific knowledge about those systems than is currently available, but more
importantly, would require the motivation to use them in the real-world context of
architectural practice. Cast in the stark terms of billable hours — time and
money — the beauty-in-proportion beliefs held by most practitioners today
appear to be rather casual.

While the parallel strands continue today, metaphysicians who search for
today’s divina proportione in the geometry and mathematics of
all manner of architectural design strategies constitute a small minority of
practitioners. New developments in computational design capabilities, however, are
opening what may be a new phase in the history of proportional systems that could
cause the parallel strands to become increasingly intertwined. Computational design
is increasingly allowing architects unprecedented control over certainty-based
proportional systems that for the past 150 years have been the domains of engineers
and other specialists.85 Parametric modeling
and other design methods are now allowing architects to solve complex problems of
design, custom fabrication of building components, and construction, while
simultaneously exploring new avenues of aesthetic expression. Formal exploration and
creative problem-solving can play variously dominant roles in these computational
design processes.86 These new certainty-based
proportional systems are essential to the design process, rather than mere
corrective appliqués like regulating lines, because to ever-increasing degrees
they are the designs. The metaphysical strand of belief-based
proportional thinking may yet be reinvigorated in this new design environment that
is so steeped in complex geometrical and mathematical operations that encourage the
production of new kinds of architectural forms that have few historical
precedents.87

Architects have always had the artistic license to explore whatever pragmatic or
metaphysical inclinations they may choose, but scholars need to strive for
objectivity in order to interpret this creative production accurately. The study of
proportional systems in the history of architecture presents a rich variety of
subject matter, very important among which is the exploration of the meanings they
communicate. Scholars have to take care to examine these meanings as historical
artifacts — not to believe them themselves, nor to create new meanings of
their own invention. Most of the following essays examine belief-based proportional
systems composed of proportions-as-ratio, or, sets of proportions as measurable,
verifiable products of artistic production. Some examine proportion-as-beauty, from
historical or historiographical viewpoints. When authors touch on
proportion-as-beauty as their own beliefs, however, such explorations constitute
forays into architectural criticism. It is my hope that the distinction between
proportion-as-ratio and proportion-as-beauty outlined in this introduction can help
readers distinguish between these various approaches to the historical phenomenon of
proportional systems, which is the subject of this volume.

2The conference had two titles: ‘Il Primo Convegno Internazionale sulle
Proporzioni nelle Arti’, and the subtitle ‘La divina
proportione’. It was held from 27–29 September 1951 as part of the
ninth Triennale di Milano, in the Palazzo dell’Arte. Published
documentation of the conference is incomplete and inconsistent. Three
publications report the contents of the conference: Wittkower (1952a), an anonymous article titled
‘Il primo convegno internazionale sulle proporzioni nelle arti’
(1952), and Cimoli and Irace (2007). Of them, ‘Il primo
convegno’ names the most speakers and other contributors, but Cimoli and
Irace publish the largest selection of texts of the papers, many of them
abridged. Out of a total of thirty-two relazioni and
communicazioni presented at the conference, Cimoli and
Irace publish twenty-five, while ‘Il primo convegno’ publishes
fourteen. The following list of participants, arranged in alphabetical order, is
derived from the latter. It retains all spellings as published, and the Italian
titles when provided, though many of the participants pursued multiple
professions: James Ackerman, Arch. Cesare Bairati, Arch. Max Bill, Luigi
Cosenza, Prof. Dekkers, Dott. Gillo Dorfles, Prof. Giusta Nicco Fasola, Scultore
Lucio Fontana, Dott. Charles Funck-Hellet, Arch. Ignazio Gardella, Prof. Matila
Ghyka, Prof. Sigfried Giedion, Mad. Carola Giedion-Welker, Prof. Hans Kayser,
Arch. Mario Labò, Le Corbusier, Arch. Carlo Mollino, Gino Levi Montalcini,
Ing. Pier Luigi Nervi, Prof. Roberto Papini, Prof. Giovanni Ricci, Prof.
Salvatore Caronia Roberti, Arch. Ernesto N. Rogers, Arch. Alfred Roth, Arch.
Piero Sanpaolesi, Pittore Gino Severini, Prof. Andreas Speiser, Prof. Eva Tea,
Dott. Adrien Turel, Pittore Georges Vantongerloo, Prof. Rudolf Wittkower and
Arch. Prof. Bruno Zevi. The following participated in a panel discussion on the
third day of the conference, but did not present papers: Arch. Annoni, Prof.
Caronia (same as Salvatore Caronia Roberti?), Ing. Enrico Castoldi, Dott.
Melino, Arch. Moretti (Luigi Moretti?), Arch. Pasqué, and Arch. Sotsas
junior. At the end of the conference the following were nominated and
unanimously elected to serve on the ‘Comitato internazionale di studio
sulle proporzioni nelle arti’ (in the order listed in ‘Il primo
convegno’): Le Corbusier (President), Arch. Phillip Johnson, Arch. Ernesto
N. Rogers, Arch. Josè Luis Sert, Prof. Andreas Speiser, Prof. Rudolf
Wittkower, Scultore Berto Lardera, Dott. Mario Melino and Signora Carla Marzoli
(‘Il primo convegno’, 1952:
119–121). According to Wittkower the name of this committee was
the ‘Comité internationale pour l’etude et l’application
des proportions dans les arts et l’industrie contemporains’, and its
purpose was to organize a second conference on proportion in the arts, to be
held in New York in 1953, though this second conference never took place (Wittkower 1952a: 55). For additional
comments and references pertaining to the Milan conference, see Mattei (2013).

3The announcement for the Leiden conference stated: ‘The purpose of this
conference is to frame a rigorous new scholarly discussion of this subject
[proportional systems in the history of architecture], and in the process, to
help define appropriate methods, standards and limits for it. The conference
will explore this subject during any period, and from both historical and
historiographical points-of-view’. For the stated purposes of the Milan
conference, see Wittkower’s opening comments quoted below.

4According to Mainstone, ‘The first recorded application of […] [the
present theories of equilibrium, deformation and strength] in structural
practice in 1742–43 may be said to mark the birth of the present art of
structural design’. This comment is in reference to a structural analysis
undertaken in those years of cracks in the dome of St. Paul’s Cathedral in
London.

5I paraphrase here the opening paragraph of Pevsner (1957b: 23 and 25): ‘A bicycle shed is a building;
Lincoln Cathedral is a piece of architecture. […] [T]he term architecture
applies only to buildings designed with a view to aesthetic appeal’,
though I expand upon Pevsner’s formulation, even beyond his concluding
sentence that the history of architecture is a history ‘primarily of
spatial expression’, by proposing that a defining quality of architecture
is its capacity to communicate iconographically.

6The prescribed proportions of streets and lots that, according to Friedman (1988), were imposed by the governing
authorities of Florentine new towns, are different than modern urban design
guidelines for building morphology in two important respects: they appear to
have been intended neither to produce practical outcomes such as health and
safety, nor to be extensive enough to meet the definition of proportional
systems presented below. At least the latter assessment also appears to apply to
the urban design guidelines for the architecture of late medieval Florence noted
by Trachtenberg (1997).

7My use of the term ‘paradigm’ here follows the fourth definition in
Oxford English Dictionary Online, which generalizes Thomas
S. Kuhn’s (1970: 10) definition to
apply to non-scientific disciplines, such as architectural history, as well as
scientific ones. The OED defines ‘paradigm’ as
‘a conceptual or methodological model underlying the theories and
practices of a science or discipline at a particular time; (hence) a generally
accepted world view’. See also Cohen (2013: 36 note 41).

8My use of the term ‘Wittkower paradigm’ is independent of
Payne’s undefined reference to ‘Wittkower’s paradigm’
(1994: 332; 2011: 46–50).

9For example, some scholars have found difficult to accept my contention that the
San Lorenzo proportional systems have no influence on the visual aesthetic value
of that basilica. See Cohen (2013; 2008).

10Both in method and result, my anecdotal observations of common opinions today
about beauty and proportional systems are similar to those reported by Claude
Perrault (1683: v–vi; 1684: 105 note 7; 1993: 49–50). On the risks of this method, see
Gerbino, ‘Were Early Modern Architects Neoplatonists?’, in this
special collection. For Wittkower’s remarks during the 1951 Milan
conference about ongoing research into Gestaltpsychologie and
the human brain, see Wittkower, ‘Finalità del convegno’ in
Cimoli and Irace (2007: 47). More
recently, Di Dio, Macaluso and Rizzolatti (2007) used functional magnetic resonance imaging (fMRI) to observe
localized regions of brain activity in test subjects who were shown sets of
proportionally manipulated photographs of canonical works of classical and
Renaissance art and asked to judge them aesthetically. Although the fMRI images
and scientific language lend the study a superficial appearance of scientific
validity, these tools are used in combination with a limited understanding of
art history and aesthetic theory. Although interdisciplinary in subject matter,
the study was apparently not conducted by an interdisciplinary research team. It
is thus compromised at the outset by the authors’ questionable,
insufficiently-supported assumptions that 1) ‘it is […] rather
implausible to maintain that beauty has no biological substrate and is merely a
conventional, experientially determined concept’ (p. 8); 2) test subjects
who are ‘naïve to art criticism’ would also be naïve to
other cultural factors that might subjectively influence their opinions about
art, or indeed about this particular experiment itself (p. 1); and 3) the golden
section ‘is considered to represent the ideal beauty’ (pp. 1 and 8).
The authors associate the latter assumption with linear overlays that they have
applied to a photograph of the Greek Doryphoros sculpture by Polykleitos,
dividing the height of this figure at the navel into two main parts, the
relative proportions of which they claim correspond to the golden section (p.
2). The authors thus bring 19th-century pseudo-scientific golden-sectionism into
a 21st-century, peer-reviewed scientific journal. These and other methodological
shortcomings provide ample reason to doubt the authors’ conclusion that
the brain activities observed by their fMRI scans affirmatively answer their
main research question of ‘whether there is an objective beauty, i.e., if
objective parameters intrinsic to works of art are able to elicit a specific
neural pattern underlying the sense of beauty in the observer’ (p. 8).
This study serves as both an example of the difficulty, and perhaps futility, of
applying scientific tools to the study of the inherently unscientific subject of
aesthetics, and a reminder that scientific training is not necessary for the
critical evaluation of such studies. For the popular dissemination of this
study, see ‘Is the Beauty of a Sculpture in the Brain of the
Beholder?’ in the 24 November issue of Science Daily
(Public Library of Science 2007). For
similarly golden-sectioned images of classical sculpture see Zeising (1854: especially 282, Fig. 188); Hagenmaier
(1949: 29); and Doczi (1985: 104–105). Le Corbusier based
his Modulor in part on the similar division of a male figure (1950: ch. 2). On the 19th-century origins
of aesthetic claims pertaining to the golden section, see van der Schoot (1998) and Frings (2002).

11Note that in his 1980 film conceived for general audiences, ‘Palladio: The
Architect and His Influence in America’, James Ackerman remarks over a
view of the interior of the central hall of Villa Poiana, ‘In
Palladio’s work, it is the refinement of proportions — a fixing of
precise ratios of length to width to depth and height — that gives one a
sense of equilibrium in and around his buildings’ (Ackerman and Terry 1980). Similarly explicit
characterizations of this belief, however, are not found in Ackerman’s
scholarly works.

12Thus, in the example of the San Lorenzo nave arcade bays, several dimensions,
when expressed in terms of the 15th-century Florentine braccio,
when separated out from the rest imply the Boethian number progression 1, 5, 9,
13, 17. These numerical relationships are not visible, but can only be
understood mentally. See Cohen (2013:
52–111; 2008).

13Cf. Herrmann’s quotation of François Blondel quoting Claude Perrault,
probably from one of the latter’s unpublished
mémoires, ‘that proportions “cannot be
seen, and therefore, cannot be the cause of a sensible effect such as the
pleasure which beauty gives us”’ (1973: 133–134).

15See also the discussion below on the limitations of correlation, as in the belief
that measurements of the proportions of buildings considered to be beautiful can
be used in the designs of new buildings in order to create similar beauty.

17I refer here to the definition of taste in Oxford English Dictionary
Online (‘taste, n.1’, entry 8a) as ‘the sense of
what is appropriate, harmonious, or beautiful; especially: discernment and
appreciation of the beautiful in nature or art; specifically: the faculty of
perceiving and enjoying what is excellent in art, literature, and the
like’. This definition merely implies the notion of sensus
communis (common sense, or communal assent), which is explicit in
the definition by Kant (1914:
169–170), quoted below (see note 25). Like Kant’s
definition, this one emphasizes taste as a judgment rather than merely a
pleasurable sensory experience.

18Cf. Oxford English Dictionary Online
(‘æsthetic’, heading A.1): ‘Of or pertaining to sensuous
perception, received by the senses’. This definition is accompanied by a
single example of usage, from the late 18th century.

19Oxford English Dictionary Online (‘æsthetic’,
heading A.2): This definition is accompanied by four examples of usage from the
19th century.

21See, for example, Billings (1840); Penrose
(1851); Henszlmann (1860); Cresy (1867); Pennethorne (1878); Thiersch (1883); Von Stegmann and Von Geymuller (1885, 1: 18); Marquand (1894); Gardner (1925); Borissavliévitch (1925); and Hambidge (1967,
a reprint of the ed. 1926, based on lectures delivered around 1916
and published in the journal The Diagonal in 1919 and 1920).
Mystically inclined studies also continued after 1949; for example, Bairati
(1952); Borissavliévitch (1952); Des Corats (1957); Funck-Hellet (1951); Jouven (1951); and
Doczi (1985). For an approach to
proportion that focuses on the concept of stability, or
résistence, see Lebrun (1807: 19–23; 1809). For an extensive but not comprehensive bibliography of
literature pertaining to proportion up to 1958, see Graf (1958). For an addendum to Graf’s bibliography, see
Borsi (1967: 119–155).

22Wittkower continued to develop and promote these theoretical generalizations in
subsequent publications; in particular, Wittkower (1953; 1960; 1962: Appendix 4, which was retained in the
1971 and 1988 editions).

23Wittkower quoting Kenneth Clark’s review of Architectural
Principles. Wittkower (1962:
Preface, v; 1971: Introduction,
n.p.) refers to this passage as his ‘intention in a
nutshell’, while Clark (1951: 65)
had merely called it a ‘result’ of Wittkower’s book.

24See Abrams (1985); Oxford English
Dictionary Online (‘æsthetic’, B.4): ‘Of or
pertaining to a late 19th-century movement in England of artists and writers who
advocated a doctrine of “art-for-art’s-sake”, also known as
the “aesthetic movement”’.

25And later on the same page: ‘Taste is then the faculty of judging a
priori of the communicability of feelings that are bound up with a
given representation (without the mediation of a concept)’. Kant’s
stipulation that taste involve judgment of ‘universally
communicable’ qualities of feelings is related to his notion of
‘Taste as a kind of sensus communis’, or
‘common sense’ (1914:
169–170). Sensus communis for Kant is not a
form of universal, metaphysical beauty (nor ‘positive beauty’,
discussed below), which would contradict the notion of judgment, but rather a
presupposition that renders judgments communicable to others. ‘The common
sense’, John Hicks explains, ‘must be assumed [in order] to be able
to agree about what the feeling is in the first place’ (2012: 111). Regarding Kant’s
stipulation that taste involve judgment ‘without the mediation of a
concept’, Hicks observes: ‘Kant […] asks us to approach
artworks formally, on their own terms, and resists interpretations that would
instrumentalize their content for the purposes of cultural critique, politics,
or philosophy itself’ (2012:
110).

26See, for example, Wittkower’s reaction to the Milan conference, discussed
in this paper.

28For classical proportions, see Gwilt (1867:
893–921, which is Book III: Practice of Architecture, Chapter
II: Principles of Proportion); and for Gothic proportions, Gwilt (1867: 963–1020, which is Book III:
Practice of Architecture, Chapter IV: Medieval Proportion, Section 6).

29Wittkower’s use of the Kunstwollen concept in this context
appears to have been inspired by Panofsky’s 1921 study of human
proportions (cited in Cohen 2013:
47).

30Cf. Le Clerc (1714: 39): ‘Par
proportion, on n’entend pas ici un rapport de raison à la maniere des
Geometres; mais une convenance de parties, fondée sur le bon goût de
l’Architecte’. For a similar use of this term in Italian from the
sixteenth century, see Van Eck’s quotation of Cosimo Bartoli in ‘The
Composto Ordinato’ in this special collection.

31For a more detailed discussion of these definitions see Cohen (2013: 21–24).

34As in, for example, a lecture by Howard Burns that I attended at the Harvard
University Graduate School of Design in 1989.

35Herrmann (1973: 138–139) notes that ‘if Perrault had not gone further
than making a distinction between positive and arbitrary qualities of beauty,
there would hardly have been much opposition’, for no one at the time
questioned the existence of positive beauty. The part of his theory that
‘was unheard of’, Herrmann continues, was his inclusion of
proportions in the category of arbitrary beauty. Note that Perrault’s
remark: ‘beauty has hardly any other foundation than
fantaisie’ appears to have been made in the context
of the immediately-preceding words ‘in buildings’, thus: ‘les
veritables regles du beau & du parfait dans les Edifices: car la Beauté
n’ayant guere d’autre fondement que la fantaisie’ (Perrault 1673: Preface, v). Indeed, eleven
years later he refers to beauty independent of buildings that has a positive
foundation and that ‘does not depend on fantaisie’:
‘une beauté qui ait un fondement tellement positif […] qui
plaisent à cause d’une proportion certaine et immuable, qui ne
dépend point de la fantaisie’ (Perrault 1684: 106 note 12). Cf. the quotations of these
passages, though with incorrect citations, in Herrmann (1973: 31, 40). So strong
was the beauty-in-proportion belief system in Perrault’s day that
Perrault’s radicalism, such as it was, according to Herrmann provides one
explanation for Perrault’s lack of influence on
contemporary architectural theory, apart from the lively debates his ideas
engendered. Herrmann (1973: 140) notes: ‘Perrault’s unorthodox
opinions were brushed aside’ by his contemporaries, who greeted them with
‘incredulity’. He furthermore notes that Perrault’s inclusion
of proportions in the class of arbitrary beauty was as ‘futile’ as
Edmund Burke’s claim, seventy years later, ‘that “proportions
are not the cause of beauty”’ (Herrmann 1973: 139, quoting Burke
1757: 91). Thus according to Herrmann’s interpretation, it would seem that
no serious challenge to the beauty-in-proportion belief system ever had a chance
of making it out of either the 17th or 18th centuries with any significant
following.

36In his Ordonnance (1683)
Perrault occasionally seems to struggle to reconcile his own distinction between
arbitrary and positive beauty in relation to architectural proportional systems,
as in his comments regarding ancient Roman proportions: ‘ancient usage is
not so much pleasing in itself as pleasing because it is linked to other
positive, natural, and reasonable beauties that make it pleasing by association,
so to speak’ (Perrault 1993: 54).
Cf. Perrault (1683: xiii): ‘les
beaux Ouvrages des Anciens […] dans lesquels aussi cette maniere ne plaist
pas tant par elle-mesme que parce qu’elle est jointe à d’autres
beautez positives, naturelles & raisonnables, laquelles, s’il faut
ainsi dire, la font aimer par compagnie’; and in his comments regarding
his own proposed proportional system for the orders that is based on the
arithmetical means of comparative measurements of ancient Roman examples:
‘even though in architecture there are, strictly speaking, no proportions
that are true in themselves, it still remains to be investigated whether it is
possible to establish probable mean proportions that are founded on positive
reasons but that do not stray too far from those that are accepted and in
current use’ (Perrault 1993:
54–55), cf. ‘& que par consequent il n’y a
point, à proprement parler, dans l’Architecture de proportions
veritables en elles-mesmes; il reste à examiner si l’on en peut
établir de probables, & de vray semblables fondées sur des raisons
positives, sans s’éloigner beaucoup des proportions reçuës
& usitées’ (Perrault 1683:
xiv). In his footnotes to his translation of Vitruvius published in
1684, Perrault similarly struggles to reconcile these two kinds of beauty, as in
his claim that even though most architects of his day believe that the
proportions of the orders presented by Vitruvius are ‘something
natural’ (‘quelque chose de naturel’), he believes these
proportions are established ‘by a consent among architects’
(‘par un consentement des Architects’), and are preferred not
because they possess positive beauty, ‘but only because these proportions
are found in works that have other kinds of positive and convincing beauty, such
as those of material and correctness of execution, and thus these proportions
are approved and appreciated even though they contain nothing positive
themselves’ (Perrault 1684: 105 note
7: ‘mais seulement parce que ces proportions se trouvoient en
des ouvrages, qui ayant d’ailleurs d’autres beautez positives &
convaincantes, telles que sont celles de la matiere & de la justesse de
l’execution, ont fait approuver & aimer la beauté de ces
proportions, bien qu’elle n’eust rien de positif’). He makes a
similar comment in a later footnote, using the term ‘veritable
beauté’ (Perrault 1684: 80 note
16). Thus Perrault seems reluctant to separate the arbitrary beauty
that he associates with architectural proportions completely from positive
beauty. Cf. Pérez-Goméz (1993:
33); and Herrmann (1973: 132).

37See Alberti’s comments: ‘But judgment[s] with regard to beauty are
not determined by opinion, but rather by an innate faculty of the mind.
[…] For there is in the forms and figures of buildings certainly a natural
excellence or perfection that excites the spirit and is immediately felt’
(Alberti 1485: IX.v, opposite fol. y:
‘Ut vero de pulchritudine iudices, non opinio, verum animis innata quaedam
ratio efficient. […] Est enim in formis profecto et figuris aedificiorum
aliquid excellens perfectumque natur. quod animum excitat evestigioque
sentiatur’). This innate or natural beauty is among the factors
contributing to Alberti’s notion of concinnitas, which
according to Van Eck (1998: 286, in reference to
De re aedificatoria 9.15) Alberti introduces
‘as a work of research, selection, and inquiry into the factors that
produce beauty, both in nature and in art’.

41Although the degree of Perrault’s influence would be difficult to
ascertain, it is notable that in 1952 Louis Hautecœur, in his
‘Préface’ in Borissavliévitch (1952: pp. 5–6), described an active, ongoing debate
in his own day about architectural proportional systems ‘between defenders
of objective beauty and defenders of subjective beauty’ (‘débat
entre défenseurs de la Beauté objective et défenseurs de la
Beauté subjective’), or terms that correspond to Perrault’s
positive and arbitrary beauty, respectively. Objective beauty, Hautecœur
notes, is ‘independent of man himself’ (‘indépendant de
l’homme même’) while subjective beauty is ‘a creation of
man’ (‘une création de l’homme’). Herrmann (1973:
150) notes that discussions of Perrault’s ‘dual nature of
beauty’ actively continued in the 18th century often under different names
than Perrault’s positive and arbitrary beauty, such as
idées as opposed to sentiments, but
always with the aim of saving absolute beauty from being undermined by the
growing notion of the relativity of taste. Consistent with both this practice
and the terms of the debate Hautecœur reports, Picon (1988: 153) interprets Perrault’s distinction between
arbitrary and positive beauty on an elemental level, as a difference between an
appeal to the senses (arbitrary) and to reason (positive):
‘L’ordonnance d’une façade peut en effet parler aux sens
ou à la raison selon son raffinement plus ou moins grand et le degré
de culture du spectateur qui la contemple’. The recent study of Di Dio,
Macaluso and Rizzolatti (2007) indicates
that interest in this debate is still active today. See note 10, above.

42Anthony Gerbino, ‘Were Early Modern Architects Neoplatonists? The Case of
François Blondel’, in this special collection.

45Such as, for example, Perrault’s argument (1683: i–v; 1993:
47–49) against any analogy between visual and musical beauty in
part because pleasing architectural proportions are more variable than pleasing
musical proportions. For Herrmann’s assessment that Perrault’s
Ordonnance was ‘not generally a success’ in
achieving what it set out to do, see Herrmann (1973: 130–189); and note
36, above.

46In contrast to the serious academic tones of Perrault and Blondel, for example,
Robert Morris cannot contain his excitement in the second volume of his
Lectures, occasionally asking forgiveness as he digresses
into ‘a kind of poetick Rhapsody’, as in his poem that begins
(italics are Morris’s): ‘Proportion! when I name
that pleasing Word, // In silent contemplative Raptures
lost, // All Nature seems to start, and say,
‘Tis here’ (Morris 1734–1736,
2: 221, also 184–186, 188–191, 200, 205–207, 209,
210–212, 215–216, 221–223).

47See for example the works listed in note 21, above.

48A phenomenon manifested in part by the Cambridge Neo-Platonists, and the
Neo-Palladian Lord Burlington group with which Robert Morris was associated (see
note 46 herein). Monod (2013); Wittkower
(1949 and 1952b: 131–132; 1962 and 1971:
150–151).

49Cf. Morris’s comment (1734–1736, 1:
Dedication): ‘Wherever Harmony resides, either in Numbers, or
Nature, it immediately strikes the imagination, by some Attractive or
Sympathizing Property’.

50Another common term that often accompanies ‘harmony’ is
‘symmetry’.

51The almost religious devotion that the golden section has sometimes inspired
since the 19th century is evident in The Society of the Golden Section
Newsletter, published in Chicago from November 1975 to May 1983. The eponymous, now-defunct
organization promoted the golden section philosophy and drawings of the
Swiss-born, American architect Abel Faidy, a self-described follower of Jay
Hambidge, Matila Ghyka and Le Corbusier. According to the society’s
executive director, Diana Faidy, in the first issue of the newsletter, the
society was formed ‘to promote specific knowledge of the golden section
and encourage employment of its disciplines, so that a new order, harmony and
symmetry may pervade the design fabric of man’s needs and the total
environment become a symphony of harmonic spatial relationships, a Unity
achieving ultimate coherence within a mathematical order’ (Faidy 1975: 1; for Abel Faidy’s
biographical information, ibid. pp. 1–2). For a similar attitude toward
the golden section, see Doczy (1985).

52I thank Judi Loach for sharing her insights on these groups with me.

53On these German and other influences, see Jean-Louis Cohen, ‘Le
Corbusier’s Modulor and the Debate on Proportion in France’ in the
present special collection. Note that the name Modulor combines the contemporary
interest in modules with the French name for golden section, section
d’or.

54A similar conference but with a more general focus was held in Paris in 1937. For
the proceedings containing 150 abstracts and papers, see Deuxième
congrès international d’esthétique et de science de
l’art (1937). In his
review of these proceedings in the American Journal of
Sociology of 1939, John T. Mueller (1939: 153) notes that while provocative as a series, ‘many of
the papers do not justify their scientific appellation’.

58James S. Ackerman and Matthew A. Cohen, ‘Proportional Systems in the
History of Architecture: A Conversation with James S. Ackerman’, in this
special collection.

59Le Corbusier uses the term ‘divina proportione’ in this manner in his
1927 essay ‘Un livre opportun’, published in this special collection
as an appendix to Jean-Louis Cohen, ‘Le Corbusier’s Modulor and the
Debate on Proportion in France’.

62Wittkower’s dismissal, based on the results of this vote, of the
beauty-in-proportion belief system as a significant cultural phenomenon of the
time is another example of his denial of pluralism in European attitudes toward
architectural proportional systems (noted above), for a 60–48 vote against
the belief that proportional systems create beauty in architecture indicates
that nearly half of the RIBA meeting participants were beauty-in-proportion
sympathizers, if not believers. The 1957 RIBA meeting had a notable parallel,
though a different outcome, in two meetings of the Académie
d’Architecture in January 1672, during which members considered the
question of ‘whether a positive rule for it [proportion] existed or
whether it was arbitrary’, and a majority voted to affirm that ‘a
positive beauty existed in architecture’ (Herrmann 1973: 32).

63To this hypothetical objection Baumgarten replies in part ‘that the
philosopher is a man amongst men and it is not good for him to think that so
great a part of human perception has nothing to do with him’.

64By associating the word ‘certainty’ with verifiable outcomes, I
distinguish it from Ackerman’s reference to ‘classical
certainty’, which implies confident yet unverifiable certainty. Thus,
belief-based proportional systems led to ‘classical certainty’
because the adherents of belief-based proportional systems were certain in their
convictions, even if those convictions were based on unverifiable and
unscientific beliefs. See James S. Ackerman and Matthew A. Cohen,
‘Proportional Systems in the History of Architecture: A Conversation with
James S. Ackerman’, in this special collection.

65The concept of ordine appears to be similar to the concept of
Gerechtigkeit (‘correct proportions’) that the
master mason Matthes Roriczer uses in his discussion of the proportions of a
Gothic pinnacle (Shelby 1977:
32–33). For a more detailed discussion of ordine,
see Cohen (2013: 270–276).

66In this definition, ‘numerical’ correspondences are the numerical
qualities of integers as revealed, for example, in number progressions, while
‘arithmetical’ correspondences are relationships between numbers
that are revealed through simple calculation. In response to the different
contexts in which they appear, I have provided here a slightly modified version
of my previous definition of proportional system, as ‘a set of
geometrical, numerical or arithmetical correspondences between important
dimensions throughout a building or major part thereof, conceived prior to the
advent of modern structural engineering in the mid-18th century, and intended by
the architect to imbue built form with desirable qualities, physical or
otherwise’ (Cohen 2013: acknowledgements
page and 22).

68See for example the comment of Emma C. Ackermann (1895: 263): ‘Whenever, in the products of art or manufacture,
there is no equal division, (symmetry), the artist or workman unconsciously
employs the proportions of the golden section. Irregular inequality and
capricious division is disagreeable to both eye and hand; and the proportion[s]
of the golden section seem to be the only acceptable ones’.

69Since even consistent, deliberate-looking proportional patterns can be
coincidental, distinguishing intentional proportions (proportions-as-ratio) from
coincidental ones may be considered one of the central challenges of the study
of architectural proportional systems. Indeed, coincidental occurrences of
highly-ordered structures must be expected in architecture, as in geometry and
mathematics. This phenomenon is aptly illuminated by Arnheim (1971: 37), who notes, ‘only in a world
based exclusively on the chance combination of independent elements is an
orderly pattern a most improbable thing to turn up; in a world replete with
systems of structural organization, orderliness is a state universally aspired
to and often brought about’. For a mathematical analysis of this
phenomenon, see Fischler (1981:
406–410).

70See note 6, above.

71For an identification of six purposes of belief-based proportional systems, none
of which may be considered practical, see Cohen (2013: 25–35).

72On the significance of 1742–1743, see note 4, above. Thus, to clarify four
of the new terms presented in this introduction, there are two kinds of
proportion: proportion-as-ratio (quantitative) and proportion-as-beauty
(qualitative); and two kinds of proportional systems: belief-based
(metaphysical) and certainty-based (scientific). Belief-based proportional
systems can be based on either proportions-as-ratio or proportions-as-beauty,
singly or in combination — these are the ‘anything goes’
proportional systems. Certainty-based proportional systems are only based on the
verifiability of proportions-as-ratio. Note that Le Corbusier worked with
engineers who used various certainty-based proportional systems (such as
engineering specifications for concrete and steel construction), in order to
ensure the structural stability and code compliance of his buildings, and that
those engineers probably ignored his belief-based Modulor.

73On my conclusion that the church prior Matteo Dolfini appears to have designed
the San Lorenzo nave arcade bay proportional system but died before he could
realize it, and that Brunelleschi inherited it from him, and modified it to
varying degrees for use in both the basilicas of San Lorenzo and Santo Spirito,
see Cohen (2013: 185–207), which
substantially expands upon Cohen (2008:
41–44).

74Whether or not Durand himself believed that his grid-based proportional system
assisted in establishing structural stability, he advocated the system in the
belief that the grid was somehow beneficial to
architecture, a belief for which there can be no scientific basis. I thus
consider it to be a belief-based proportional system.

75Cf. Pevsner’s remarks in note 5, above.

76‘su solo albedrio […] por ciertas lineas ortogonales lo hacen y se
osan encomendar a ello’. This comment was made in the context of
determining the proportions of a Gothic buttress. On the ineffectiveness of
proportion-as-ratio in establishing structural stability, see also Curti,
‘Canons of Proportion’, in this special collection.

77Anthony Gerbino, ‘Were Early Modern Architects Neoplatonists? The Case of
François Blondel’ and Konrad Ottenheym, ‘Dutch
Seventeenth-Century Proportional Design Systems’, both in this special
collection.

78See Da Vignola’s comment (1562: Prefazione,
n.p.): ‘just as every one of our senses delights in this
proportion, and the displeasing things fall outside of it, as the music
theorists have well and judiciously proven in their science’
(‘quanto ogni nostro senso si compiaccia in questa proporzione, e le cose
spiacevoli essere fuori di quella, come ben provano li musici nella loro scienza
sensatamente’); and Palladio’s similar comment: ‘just as the
proportions of voices are harmony to the ears, so those of measurement are
harmony to the eyes, which according to their habit delights to a great degree,
without it being known why, apart from those who study to know the reasons of
things’ (‘perciochè, secondo che le proportioni delle voci sono
armonia delle orecchie così quelle delle misure sono armonia degli occhi
nostri, la quale secondo il suo costume sommamente diletta, senza sapersi il
perchè, fuori che da quelli che studiano di sapere le ragioni delle
cose’, as transcribed in Zorzi 1967:
88; and with slight differences in Palladio 1988: 123). Cf. Palladio’s similar comments in
Palladio (1570: iv). For additional
discussion, see Cohen (2013:
33–35).

79An example of one of these myriad factors is cultural association, as when
government buildings from the Fascist period in Italy are considered by some to
be unsightly due to their historical associations, their classical
proportions-as-ratio proportions notwithstanding.

80Regarding Alberti’s belief in causation, see his comment that those
‘numbers by means of which the harmony of voices is very pleasing to the
ear, are the same numbers that please the eyes and the spirit’ (‘Hi
quidem numeri, per quos fiat ut vocum illa concinnitas auribus gratissima
reddatur, hidem ipsi numeri perficiunt, ut oculi animusque voluptate mirifica
compleantur’) (Alberti 1415: IX.v, yii [verso]). I thank Darrin Griechen
for suggesting the causation/correlation couplet as a tool for discussing
certain qualities of proportional systems.

81In the pre-engineering period, the notions of structural stability, beauty and
ordine all overlapped. Since structural stability
constituted such a pressing, pragmatic need that at best could be satisfied only
some of the time, primarily through the experience and skill of the builders,
availability of high-quality materials, and luck, the then-interrelated notions
of structural stability, beauty and ordine can in this context
be considered to have had pragmatic intentions. Proportional systems did not
contribute to structural successes, but most architects and builders probably
thought they did. The motivations of architects who have used belief-based
proportional systems during the post-engineering period, conversely, may be
assumed to have been more mystically oriented, because with the availability of
modern structural engineering to ensure structural stability, the motivations
for using such proportional systems cannot have been pragmatic.

82My observation of parallel strands of belief that linked groups of thinkers
across the centuries, one strand more metaphysical and the other more pragmatic
in relation to each other, is consistent with Wittkower’s observation
(1949 and 1952: 127; 1962 and 1971: 145), for example, that while Briseux, in his
Traité du Beau essentiel dans les arts of 1752,
defends Blondel’s principles against Perrault, Briseux nevertheless
reveals a ‘shift of emphasis from universally valid to psychologically
conditioned standards’. According to my interpretation, however, this
shift of emphasis occurred within the metaphysical strand, and this strand
continued into later centuries, by contrast with Wittkower’s
interpretation that Briseux’s views represented a dying mode of thinking.
Both Blondel and Briseux may be considered to have contributed to the
metaphysical strand, each in his own way.

83See Ackerman’s interpretation of the Milan conference as ‘the end of
the mystical phase’ of the study of proportional systems, and the
beginning of the ‘effort to set it onto reliable, academic, practical
grounds’ (Cohen and Ackerman, ‘Proportional Systems in the History
of Architecture: A Conversation with James S. Ackerman’, in this special
collection).

85See Robin Evans’s (1995) analysis of
the shift in the locus of geometrical innovation from architecture to
engineering, where it has remained until very recently.

86Thus in their survey of recent developments in the use of mathematics in
architecture, Jane Burry and Mark Burry (2010:
13) note of the projects they present: ‘there is a natural
division between those in which the primary mathematical constituent is an idea,
and those where mathematics is first and foremost positioned as a
problem-solver. In some, the two roles are balanced or combined, and in all, the
mathematical idea or problem-solver is also instrumental in the design process
and to the form of the architectural outcome’.

87Two excellent sources for sampling the great variety of attitudes and approaches
to the uses of geometry and mathematics in architecture today are the journals
Architectural Design (London) and the Nexus Network
Journal (Turin).

References

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